Solution Of Heat Transfer Equation Research Articles

Thermo-large deflection coupled dynamic characteristics of rotating thickness-varying plates subjected to thermal shock

Thermal shock load has an important influence on dynamic characteristics of blades in turbomachines. Due to higher level operating thermal shock for improving the efficiency, the damage to the blades has greatly increased. Considering the major effect of variable thickness and large deflection on thermal dynamic behavior of blades, their thermo-large deflection-variable thickness coupled dynamic characteristics are unknown. This paper develops a thermo-large deflection coupled semi-analytical model based on a rotating variable thickness plate under thermal shock. Geometric nonlinear governing equation and an analytical solution of heat transfer equation are derived. Numerical examples of natural frequencies are investigated to validate the proposed model. The non-uniform temperature distribution features of the rotating variable thickness plate are found. Besides, when the thermal shock is serious, the large deflection theory must be considered to analyze thermo-large deflection coupled dynamic characteristics. Based on theoretical and experimental results, deformation characteristic of the cantilever plate subjected to thermal shock is obtained, which is helpful to find the optimal thickness distribution. Finally, the effect of variable thickness on thermo-large deflection coupled dynamics of the rotating plate is demonstrated. The vibration induced by thermal shock can be reduced by thickening the plate on its free side or both fixed–free sides. However, when thickening other positions, the vibration is either intensified or unchanged. Results are helpful to reduce blade’s vibrations under thermal shock.

Exact solution for heat transport of Newtonian fluid with quadratic order thermal slip in a porous medium

In this communication, an analytical solution for the thermal transfer of Newtonian fluid flow with quadratic order thermal and velocity slips is presented for the first time. The flow of a Newtonian fluid over a stretching sheet which is embedded in a porous medium is considered. Karniadakis and Beskok’s quadratic order slip boundary conditions are taking into account. A closed form of analytical solution of momentum equation is used to derive the analytical solution of heat transfer equation in terms of confluent hyper-geometric function with quadratic order thermal slip boundary condition. Accuracy of present results is assured with the numerical solution obtained by Iterative Power Series method with shooting technique. The impacts of porous medium parameter, tangential momentum accommodation coefficient, energy accommodation coefficient on velocity and temperature profiles, skin friction coefficient and reduced Nusselt number are discussed. The Nusselt number increases with the higher estimations of tangential momentum and energy accommodation coefficients.

ГЕОТЕРМИЯ И ОЦЕНКА НЕФТЕГАЗОВОГО ПОТЕНЦИАЛА БУРЕИНСКОГО БАССЕЙНА (ДАЛЬНИЙ ВОСТОК РОССИИ)

Актуальность. Буреинский осадочный бассейн – наиболее интересная в нефтегазоносном отношении межгорная впадина слабоизученной материковой части Дальнего Востока России. Положительная оценка в отношении перспектив обнаружения в отложениях бассейна месторождений нефти и газа обоснована результатами ранее проведённых производственных и научных работ. Цель: дальнейшая аргументация перспектив нефтегазоносности верхнеюрско-нижнемеловых отложений Буреинского бассейна на основе применения отечественного программного обеспечения бассейнового моделирования TeploDialog, обладающего оригинальными особенностями. Объект: осадочные комплексы средне-верхнеюрский, верхнеюрско-нижнемеловой и меловой Кындалского грабена, наиболее хорошо изученной структуры в составе Буреинского краевого прогиба. Ранее проведенные исследования показали, что в процесс нефтегазообразования вовлечён весь юрско-меловой разрез Буреинского бассейна. Методы. В комплексе ТeploDialog реализован метод палеотемпературного моделирования, основанный на численном решении уравнения теплопроводности горизонтально-слоистого твердого тела с подвижной верхней границей. В математическую модель непосредственно включены климатический вековой ход температур на земной поверхности, как краевое условие, и палеотемпературы из определений отражательной способности витринита, как «наблюденные». Метод не требует априорных сведений о природе и величинах глубинного теплового потока, поток определяется решением обратной задачи геотермии в рамках параметрического описания седиментационной истории и истории теплофизических свойств осадочной толщи. Результаты. Установлено, что геотермия грабена, начиная с юрского времени, благоприятствовала процессам генерации жидких углеводородов в породах осадочного чехла, которые могли начаться около 107 млн лет назад, в середине альба. Наиболее интенсивно нефтегенерация происходила в период 107–40 млн лет назад, до середины эоцена. Температурные условия генерации жидких углеводородов сохраняются для талынджанского очага. Наибольшая продолжительность температурных периодов, благоприятных для газообразования, наблюдается в чемчукинском очаге. Условия генерации газа сохраняются для ургальского и чагдомынского очагов. Результаты, полученные с использованием отечественной системы TeploDialog и ранее полученные в системе PetroMod, во многом совпадают. Названные системы характеризуются разными подходами к рассмотрению тектоно-седиментационных и термических условий генерации УВ. Поэтому результаты настоящих исследований являются дополнительной аргументацией положительных перспектив нефтегазоносности верхнеюрско-нижнемеловых отложений Буреинского бассейна.

Impact of Liquid Medium on Laser Ablation Mechanism: Surface Heating and Cooling

Stainless steel target was ablated under both of Hexane and water to examine the effect of liquid on the ablation mechanism in terms of surface heating and cooling. Experimentally, a narrow, a smooth and a deep ablation is observed under water. In addition, the pit due to the collapse of cavitation bubble under water is insignificant as compared with Hexane. The focused beam on the target under liquid was estimated from measuring the crater’s diameter from the optical image. The difference in focused beam sizes is attributed mainly to the refractive index of the liquid. Heat transfer equation is solved to estimate the temperature profile on the solid surface for a single pulse. By considering the estimated beam size under liquid, the solution of heat transfer equation suggests for the first time rapid heating of the target under water, but doesn’t confirm the rapid cooling rate under Hexane. The smooth ablation in water is attributed to the more confined plasma.

Finite Element Analysis to Determine Residual Stress in Electron Beam Welding of CuCrZr alloy Plates and Experimental Validation

The study on determining residual stresses in welding process through finite element (FE) analysis could receive much attention in the last decades. The prediction of residual stresses and temperature distribution is important for many industries, as it can ensure more efficient mechanical design. Electron beam welding (EBW) of copper chromium zirconium (CuCrZr) alloy was carried out in the bead-on-plate configuration. The temperature fields and stress distributions in the welded joint during the welding process were numerically simulated and experimentally verified. 3D finite element models were developed using the solution of heat transfer equations in Ansys mechanical APDL. Here, conical heat source was used for simulation. The temperature distribution was obtained, and stress analysis was carried out with the developed model using the temperature dependent material properties for CuCrZr.The temperature distribution was symmetric along the weld center, and the thermal stress also appeared to be in symmetric distribution. The predicted results were verified through real experiments, and a good agreement between them was achieved.

3D Thermal Analysis for CFETR Pre-Superheated Water-Cooled Blanket

Initial China Fusion Engineering Test Reactor (CFETR), blanket design, includes water-cooled ceramic breeder (WCCB) blanket operating in pre-superheated regime. This condition allows efficient cooling; however it requires accurate control and analysis to avoid zones with excessive heat flux. Analysis of the coolant flow and heat transfer in CFETR Pre-Superheated Blanket was performed using ANSYS CFX and included: 3D coolant flow analysis, external volumetric and surface heating effect, and two-phase wall boiling. ASIPP CAD Model imported directly into ANSYS Workbench Design Modeler as a STEP file. Fluid volume is created using Design Modeler Fill operation, and converting Inlet and Outlet surfaces. Meshing was performed using CFX method available within the framework of the ANSYS mesh generator. Application of tetrahedral elements for meshing of the internal regions allowed automatic mesh generation. Advanced sizing functions were used with automatic mesh inflation depending on wall proximity and curvature. Conjugated heat transfer analysis was performed including solution of heat transfer equations in solid and liquid parts, and solution of the flow equations in the liquid parts. Coolant flow in that was assumed turbulent and was resolved using Reynolds averaged Navier-Stokes equations with Shear Stress Transport turbulence model. RPI model for wall driven boiling is used. Inhomogeneous two-phase flow is resolved solving two sets of momentum and energy equations for liquid and steam. Results showed ability of the model to simulate two-phase boiling flow in complex configuration.

Conditions for reducing spherical thermal aberrations in transversally pumped YAG rods

Temperature dependences of the thermal conductivity and thermo-optic coefficient are often neglected when thermal lensing in laser crystals is investigated, though their influence is very significant. In this paper, the general solution of heat transfer equation with temperature dependent thermal conductivity is found in an integral form, which is very convenient for analytical and numerical analysis. Using this solution, the possibility to eliminate spherical aberration by a proper choice of the pump parabolicity parameter is investigated in detail. The inaccuracies in the definition of optical path differences used in a few previous works for the case of temperature dependent thermo-optic coefficient are explained. It is shown that the use of a correct definition increases the value of the parabolic coefficient at which the elimination of spherical aberrations may be realized as compared with the original work of Hodgson and Weber. It is also found that nearly the same shapes of pumping are required for elimination of spherical aberrations for both radial and tangential polarization.

Investigation of temperature and residual stresses field of submerged arc welding by finite element method and experiments

This article reports on a numerical and experimental investigation to understand and improve computer methods in application of the Goldak model for predicting thermal distribution in submerged arc welding (SAW) of APIX65 pipeline steel. Accurate prediction of the thermal cycle and residual stresses will enable control of the fusion zone geometry, microstructure, and mechanical properties of the SAW joint. In this study, a new Goldak heat source distribution model for SAW is presented first. Both 2D and 3D finite element models are developed using the solution of heat transfer equations in ABAQUS Standard implicit. The obtained results proved that the 2D axi-symmetric model can be effectively employed to simulate the thermal cycles and the welding residual stresses for the test steel. As compared to the 3D analysis, the 2D model significantly reduced the time and cost of the FE computation. The numerical accuracy of the predicted fusion zone geometry is compared to the experimentally obtained values for bead-on-plate welds. The predictions given by the present model were found to be in good agreement with experimental measurements.

Modelling and Analysis of Phase Transformations and Stresses in Laser Welding Process / Modelowanie I Analiza Przemian Fazowych I Naprężeń W Procesie Spawania Laserowego

The work concerns the numerical modelling of structural composition and stress state in steel elements welded by a laser beam. The temperature field in butt welded joint is obtained from the solution of heat transfer equation with convective term. The heat source model is developed. Latent heat of solid-liquid and liquid-gas transformations as well as latent heats of phase transformations in solid state are taken into account in the algorithm of thermal phenomena. The kinetics of phase transformations in the solid state and volume fractions of formed structures are determined using classical formulas as well as Continuous-Heating-Transformation (CHT) diagram and Continuous-Cooling-Transformation (CCT) diagram during welding. Models of phase transformations take into account the influence of thermal cycle parameters on the kinetics of phase transformations during welding. Temporary and residual stress is obtained on the basis of the solution of mechanical equilibrium equations in a rate form. Plastic strain is determined using non-isothermal plastic flow with isotropic reinforcement, obeying Huber-Misses plasticity condition. In addition to thermal and plastic strains, the model takes into account structural strain and transformation plasticity. Changing with temperature and structural composition thermophysical parameters are included into constitutive relations. Results of the prediction of structural composition and stress state in laser butt weld joint are presented.

Numerical Analysis of Coolant Flow and Heat Transfer in ITER Diagnostic First Wall

Numerical simulations of the ITER Diagnostic First Wall (DFW) were performed using ANSYS workbench. During operation DFW will include solid main body as well as liquid coolant. Thus thermal and hydraulic analysis of the DFW was performed using conjugated heat transfer approach, in which heat transfer was resolved in both solid and liquid parts, and simultaneously fluid dynamics analysis was performed only in the liquid part. This approach includes interface between solid and liquid part of the system. Analysis was performed using ANSYS CFX software. CFX software allows solution of heat transfer equations in solid and liquid part, and solution of the flow equations in the liquid part. Coolant flow in the DFW was assumed turbulent and was resolved using Reynolds averaged Navier-Stokes equations with Shear Stress Transport turbulence model. Meshing was performed using CFX method available within ANSYS. The data cloud for thermal loading consisting of volumetric heating and surface heating was imported into CFX. Volumetric heating source was generated using Attila software. Surface heating was obtained using radiation heat transfer analysis.Results allowed to identify areas of excessive heating. Proposals for cooling channel relocation were made. Additional suggestions were made to improve hydraulic performance of the cooling system.

Analytical and Numerical investigation of natural convection in a heated cylinder using Homotopy Perturbation Method

Homotopy Perturbation Method (HPM) has been applied to solve a nonlinear heat transfer problem. Natural convection around an isothermal horizontal cylinder was studied. Heat transfer coefficient and specific heat coefficient were assumed to be dependent on temperature. Outcomes were compared with solution of heat transfer equation with constant properties. Solutions of HPM were compared with numerical results for different cases, and variation of Nusselt number was obtained and investigated.

Application of variational iteration method to nonlinear heat transfer equations using He's polynomials

PurposeThe purpose of this paper is to solve an unsteady nonlinear convective‐radiative equation and a nonlinear convective‐radiative‐conduction equation containing two small parameters of ε1 and ε2 by variational homotopy perturbation method.Design/methodology/approachThe heat transfer equations are described. The variational homotopy perturbation method as a powerful method for solving linear and nonlinear equations is applied to find the solutions of our model equations.FindingsThe solutions of heat transfer equations are calculated in the form of convergent series with easily computable components. Two examples are solved as illustrations, using symbolic computation.Originality/valueThe results show that the suggested method is easy to implement and has high level of accuracy. The method introduces a reliable tool for solving many linear and nonlinear differential equations.

Approximate solution of non-Newtonian viscoelastic fluid flow on a turbine disc for cooling purposes by using Adomian decomposition method

In this work, a powerful semi-analytical method, called Adomian decomposition method (ADM) is introduced to obtain the exact solution of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall which is useful for turbine cooling applications. This method is employed to obtain the expressions for velocity and temperature fields. Then, the influence of the two dimensionless numbers: the Prandtl number (Pr) and the Reynolds number (Re) for the dynamic forces have been considered. Comparisons are made between the numerical method (NM) solution, Homotopy perturbation method (HPM), Variation iteration method (VIM) and the results of Adomian decomposition method (ADM). The results reveal that this method is very effective and simple and can be applied for other nonlinear problems.

Thermal modeling of a concentrator pipe composed with direct steam generation

Deflections along the pipes of solar concentrators used for direct steam generation in thermosolar plants lead to some structural problems, due to temperature gradients in radial and angular directions. In this work, a thermal analysis of an absorber copper-steel pipe is made, based on the solution of heat transfer equations in stable and transient state, for the solution of model was used the finite differences method. Radial and angular temperature distributions are presented, as well as the instant at which the system reaches the stationary state under the studied conditions. The considered absorber pipe is a composed walls system of 20% Cu and 80% Steel, constituted by commercial copper pipe of type k with 61.849 and 66.675 mm of internal and external diameter respectively; and a steel pipe of schedule 160 with 66.7 mm of internal diameter and 88.9 mm of external diameter. The properties in the composed wall are favorable because achieves an isothermal state with a maximum gradient of 50°C/m. Therefore, the composed pipe proposed offers greater resistance to the deflection provoked by the direct steam generation.

Dissipation Effects on MHD Nonlinear Flow and Heat Transfer Past a Porous Surface with Prescribed Heat Flux

Viscous and Joule dissipation effects are considered on MHD nonlinear flow and heat transfer past a stretching porous surface embedded in a porous medium under a transverse magnetic field. Analytical solutions of highly nonlinear momentum equation and confluent hypergeometric similarity solution of heat transfer equations in the case when the plate stretches with velocity varying linearly with distance are obtained. The effect of various parameters like suction parameter, Prandtl number, Magnetic parameter, and Eckert number entering into the velocity field, temperature distribution and skin friction co-efficient at the wall are discussed with the aid of graphs.

Homotopy Analysis Method for the heat transfer of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall

In this article, a powerful analytical method, called the Homotopy Analysis Method (HAM) is introduced to obtain the exact solutions of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications. The HAM is employed to obtain the expressions for velocity and temperature fields. Tables are presented for various parameters on the velocity and temperature fields. These results are compared with the solutions which are obtained by Numerical Methods (NM). Also the convergence of the obtained HAM solution is discussed explicitly. These comparisons show that this analytical method is strongly powerful to solve nonlinear problems arising in heat transfer.

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source

In this paper, visco-elastic boundary layer flow and heat transfer over a stretching sheet in presence of viscous dissipation and non-uniform heat source have been discussed. Analytical solutions of highly non-linear momentum equation and confluent hypergeometric similarity solution of heat transfer equations are obtained. Here two types of different heating processes are considered namely (i) prescribed surface temperature (PST) and (ii) prescribed wall heat flux (PHF). The effect of various parameters like visco-elastic parameter, Eckert number, Prandtl number, and non-uniform heat source/sink parameter on temperature distribution are analyzed and effect of all these parameters on wall temperature gradient and wall temperature are tabulated and discussed.

Temperature evolution during stereolithography building with a commercial epoxy resin

AbstractIn this study, the temperature evolution during stereolithography (SL) processing of a commercial epoxy resin is experimentally measured and numerically simulated. Experimental tests were performed in a SL equipment to evaluate the temperature increase during laser‐activated photopolymerization. Temperatures on the resin surface were measured with a thermal video system in both static and moving laser experiments. For the moving laser experiments, the effect of the energy dose was tested by using different velocities of the scanning laser. The experimental results were compared with numerical model prediction obtained by numerical solution of heat transfer equations coupled with an original mathematical model developed for cationic photopolymerization kinetics. The results obtained from numerical simulation were in good agreement with experimental data for the scan performed at the lower energy dose. The process model describes both the temperature increase and the evolution of chemical reaction, providing information about the penetration depth and the cured linewidth. POLYM. ENG. SCI. 46:493–502, 2006. © 2006 Society of Plastics Engineers

Thermal and stress analysis of a sheet metal in welding

A cantilever assembly subjected to heating at its fixed end, which resembles the welding of a sheet metal is considered. Temperature and stress fields are computed during the heating process. A control volume approach is introduced for the numerical solution of heat transfer equations while the finite element method is adopted for stress field predictions. It is found that the temperature distribution in the transverse direction does not vary considerably, but varies significantly in the longitudinal direction. The temporal change of temperature gradient induces stresses in the substrate material. However, the maximum magnitude of the von Mises stress is less than the yield strength of the substrate material.

Kinetics of Steel Scrap Melting in Molten Fe-C Bath

The melting phenomena of steel scrap in molten steel bath has been investigated by several workers. Their works are reviewed in the present paper. The phenomena are, in general, so complicated that both heat and mass transfer take place simultaneously. Many workers made small scale experiments in such condition that only mass transfer was taken into consideration. The results were summarized in the form of dimensionless correlation among Sh, Gr. Re, Sc and J D . Only one paper reported the experiment in which pure heat transfer was taken into consideration. The overall heat transfer coefficient was estimated by fitting numerical solution of heat transfer equations to the results of immersion experiments. The simultaneous heat and mass transfer were analyzed by mathematical model in relation to BOF converter operation. Some workers made experiments and numerical solution of mathematical model to obtain the heat and mass transfer coefficients in pilot plant converter and commercial converter.